382 Sir W. Thomson on Periodic Motion. 



to move exactly perpendicularly to it. If the next time their 

 line passes through S they are again moving perpendicularly 

 to ME, their motion relatively to S I is rigorously periodic. 

 This we see by considering that if both motions are reversed 

 at any instant, M and E will exactly retrace their paths ; and 

 if such a reversal is made at an instant of perpendicularly 

 crossing the line S T, the retraced paths are similar to the 

 direct paths which are traced when there is no reversal. 



19. Hence if the three bodies be given in line, SME, we 

 secure rigorous periodicity of their motion if we project them 

 in contrary directions perpendicular to this line with exactly 

 such velocities that the next time M E is again in line with S, 

 now S E M, their directions of motion are again perpendicular 

 to E M. The problem of doing this has three solutions ; in one 

 of which the velocities of projection are so great that M and 

 E are carried far away from one another, in opposite direc- 

 tions round the Sun till they again come near one another 

 and in line on the far side of the Sun. Excluding this case 

 we have certainly only two solutions left. In these I describes 

 exceedingly nearly a circle round the Sun ; while M and E 

 move relatively to the point I and the line I S, somewhat 

 approximately in circles, but to a second approximation in the 

 ellipses corresponding to the lunar perturbation called the 

 variation, and quite rigorously in two constant similar closed 

 curves each differing very little from the variational ellipse. 

 The centre of the variational ellipse is at I : its major axis is 

 perpendicular to S I and exceeds the minor axis by approxi- 

 mately 1/179*6, being the square of the ratio (1/13*4) of the 

 angular velocity of S I to the angular velocity of ME, each 

 relative to an absolutely fixed direction. There are two solu- 

 tions of this kind, in one of which (as in the actual case of 

 Earth and Moon) E M turns samewards as, in the other 

 contrary- wards to, S E. 



20. If M E were two or three times as great as it is when 

 the three bodies are in line, SME, and other dimensions the 

 same, we should still have a solution for periodicity corre- 

 sponding to that of § 19, but with the orbital curves of M and 

 S round I differing very largely from circles and largely from 

 ellipses. When M E exceeds a certain limit, this kind of solution 

 becomes impossible. It would be not wholly uninteresting to 

 follow the character of the orbital curves round I for increas- 

 ing magnitudes of ME until they are lost. The solution 

 referred to and rejected in § 19 is still available and becomes 

 now more interesting, but not so interesting as the correspon- 

 ding solution in which M and E, now two planets, are pro- 

 jected so as to revolve in the same directions round the Sun. 



